The multiplicative ART (MART) is an iterative algorithm closely re- lated to the ART. It also was devised to obtain tomographic images, but, like ART, applies more generally; MART applies to non-negative systems of linear equationsAx=bfor which the bi are positive, the Aij are non-
negative, and the solution x we seek is to have nonnegative entries. It is not so easy to see the relation between ART and MART if we look at the most general formulation of MART. For that reason, we begin with a sim- pler case, transmission tomographic imaging, in which the relation is most clearly apparent.
4.5.1
A Special Case of MART
We begin by considering the application of MART to the transmission tomography problem. For i = 1, ..., I, let Li be the set of pixel indices j
for which thej-th pixel intersects thei-th line segment, and let|Li|be the
cardinality of the setLi. Let Aij = 1 forj in Li, andAij = 0 otherwise.
Withi=k(modI) + 1, the iterative step of the ART algorithm is
xkj+1 =xkj + 1
|Li|
(bi−(Axk)i), (4.13)
forj inLi, and
if j is not in Li. In each step of ART, we take the error, bi −(Axk)i,
associated with the current xk and the i-th equation, and distribute it equally over each of the pixels that intersectsLi.
Suppose, now, that eachbi is positive, and we know in advance that the
desired image we wish to reconstruct must be nonnegative. We can begin withx0>0, but as we compute the ART steps, we may lose nonnegativity.
One way to avoid this loss is to correct the current xk multiplicatively,
rather than additively, as in ART. This leads to the multiplicative ART (MART).
The MART, in this case, has the iterative step
xkj+1=xkj bi
(Axk) i
, (4.15)
for thosej inLi, and
xkj+1=xkj, (4.16)
otherwise. Therefore, we can write the iterative step as
xkj+1=xkj bi (Axk) i Aij . (4.17)
4.5.2
The MART in the General Case
Taking the entries of the matrix Ato be either one or zero, depending on whether or not thej-th pixel is in the set Li, is too crude. The lineLi
may just clip a corner of one pixel, but pass through the center of another. Surely, it makes more sense to letAij be the length of the intersection of
line Li with the j-th pixel, or, perhaps, this length divided by the length
of the diagonal of the pixel. It may also be more realistic to consider a strip, instead of a line. Other modifications toAij may be made, in order
to better describe the physics of the situation. Finally, all we can be sure of is thatAij will be nonnegative, for eachi andj. In such cases, what is
the proper form for the MART?
The MART, which can be applied only to nonnegative systems, is a sequential, or row-action, method that uses one equation only at each step of the iteration.
Algorithm 4.2 (MART) Let x0 be a positive vector. For k = 0,1, ..., andi=k(modI) + 1, having foundxk definexk+1 by
xkj+1=xkj bi (Axk) i m−i1Aij , (4.18) wheremi= max{Aij|j= 1,2, ..., J}.
Some treatments of MART leave out the mi, but require only that the
entries of Ahave been rescaled so that Aij ≤1 for alli and j. Themi is
important, however, in accelerating the convergence of MART.
Notice that we can writexkj+1as a weighted geometric mean ofxkj and
xk j b i (Axk)i : xkj+1=xkj 1−m−1 i Aij xkj bi (Axk) i m−i1Aij . (4.19)
This will help to motivate the EM-MART.
4.5.3
Cross-Entropy
For a > 0 and b > 0, let the cross-entropy or Kullback-Leibler (KL) distance fromato bbe
KL(a, b) =aloga
b +b−a, (4.20)
with KL(a,0) = +∞, and KL(0, b) = b. Extend to nonnegative vectors coordinate-wise, so that KL(x, z) = J X j=1 KL(xj, zj). (4.21)
Unlike the Euclidean distance, the KL distance is not symmetric;
KL(Ax, b) and KL(b, Ax) are distinct, and we can obtain different ap- proximate solutions of Ax = b by minimizing these two distances with respect to non-negativex.
4.5.4
Convergence of MART
In the consistent case, by which we mean that Ax=bhas nonnegative solutions, we have the following convergence theorem for MART.
Theorem 4.2 In the consistent case, the MART converges to the unique nonnegative solution ofb=Ax for which the distancePJ
j=1KL(xj, x 0
j)is minimized.
If the starting vector x0 is the vector whose entries are all one, then the MART converges to the solution that maximizes theShannon entropy,
SE(x) =
J
X
j=1
As with ART, the speed of convergence is greatly affected by the order- ing of the equations, converging most slowly when consecutive equations correspond to nearly parallel hyperplanes.
Open Question: When there are no nonnegative solutions, MART does not converge to a single vector, but, like ART, is always observed to produce a limit cycle of vectors. Unlike ART, there is no proof of the existence of a limit cycle for MART. Is there such a proof?